Playing Penrose's Tile Game
David Austin
September 21, 2006

In this talk, I will describe so-called Penrose tilings. Everyone is familiar with tiles; they are all about us on the floors of our kitchens and bathrooms. Most of these tiles have a simple shape, such as a square, and the floor is tiled by repeatedly laying down one tile next to the one before it in a very orderly way. The overall pattern, which can be pleasant to look at, is created by the repetition of a single, smaller pattern.

Penrose tiles are a collection of tiles that can also be put together to cover an area like your kitchen floor. However, the patterns they form are not created by repeating a single pattern, and this is what makes them so interesting to mathematicians and others. Because they are not formed by repetition, there is a disorderliness to the patterns. Yet at the same time, there is a principle, which will be explained in this talk, that leads to a new type of order and allows us to investigate these patterns.

Roger Penrose discovered these tiles in the mid 1970's while doodling during a visit to a relative in the hospital. About ten years later, a radically new phenomenon was observed in crystallography; after further investigation, it was found that Penrose's tiles provided an explanation of this phenomenon. This illustrates something fundamental about the nature of mathematics: new mathematics is usually created out of simple curiosity, yet it often provides clear, elegant explanations of important features of our natural world.

The Chaos Game and Fractal Images
Bob Devaney
October 19, 2006

In this lecture we will describe some of the beautiful images that arise from the "Chaos Game." We will show how the simple steps of this game produce, when iterated millions of times, the intricate images known as fractals. We will describe some of the applications of this technique used in data compression as well as in Hollywood. We will also challenge students present to "Beat the Professor" at the chaos game and maybe win his computer.

Fibonacci's Garden
Matt Boelkins
February 8, 2007

By 300 BC, Euclid and other Greek mathematicians were aware of a number with special properties linked to proportion in geometric figures. Specially divided line segments, aesthetic rectangles, and regular pentagons all exhibited an amazing number that later came to be known as "the Golden Ratio." The Golden Ratio has since made many appearances in surprising places in mathematics, including rather recently in symmetries in Penrose tilings, and has also manifested itself in curious ways in art, architecture, and the natural world.

Around 1200 AD, Leonardo of Pisa (known to us today as Fibonacci) began experimenting with a sequence of numbers that has since come to bear his name. The list of numbers generated by starting with 0 and 1 and then adding the two previous numbers to find the next term results in

is called "the Fibonacci Sequence." This collection of numbers has been discovered to have a seemingly unlimited list of interesting properties that fascinate mathematicians to this day. Like the Golden Ratio, Fibonacci numbers arise naturally in some startling places. One example is seen in pine cones, where the numbers of spirals exhibited on the pine cone in opposing directions normally turn out to be consecutive Fibonacci numbers.

Perhaps even more remarkably, the Golden Ratio and the Fibonacci Sequence are inextricably linked with each other. After an introduction to some of the history and mathematical ideas surrounding each of these concepts separately, we will explore how the development of seeds in flowers demonstrates some of these connections between the Golden Ratio and the Fibonacci Sequence.